The population of a city in 2005 was 18,000. By 2010, the city's population had grown to 45,000. Economists
have determined that the population growth follows an exponential model. If they are correct, what is the
projected population for 2015?
To determine the projected population for 2015 using an exponential model, we can use the formula:
P(t) = P0 * e^(rt)
Where:
P(t) = population at time t
P0 = initial population
e = Euler's number (approximately 2.71828)
r = growth rate
t = time elapsed
Given:
P0 = 18,000
P(2010) = 45,000
t = 2010 - 2005 = 5 years
First, we need to find the growth rate (r) using the initial and final population values:
45,000 = 18,000 * e^(5r)
45,000 / 18,000 = e^(5r)
2.5 = e^(5r)
Taking the natural logarithm (ln) of both sides to solve for r:
ln(2.5) = ln(e^(5r))
ln(2.5) = 5r
r ≈ ln(2.5) / 5
r ≈ 0.406
Now, we can use the growth rate to find the projected population for 2015 (t = 10 years):
P(2015) = 18,000 * e^(0.406*10)
P(2015) = 18,000 * e^4.06
P(2015) ≈ 18,000 * 58.223
P(2015) ≈ 1,049,014
Therefore, the projected population for 2015 would be approximately 1,049,014.